The reverse Hoelder inequality for power means

摘要

For a function φ non-negative on the interval [0, 1], the power mean of order α ≠ 0 is defined by the equality M_(αφ)(t) = (1/t ∫_0~tφ~α(u) du)~(1/α), 0 < t ≤ 1. We consider the class RH~(～α,β)(B) of functions φ satisfying the reverse Hoelder inequality M_(βφ) ≤ B·M_(αφ) at some α < β, α·β≠ 0, B > 1. The sharp estimates for the summability exponents of the compositions of power means are established. As a result, we determine the properties of self-improvement of the RH~(～α,β)(B). Summability exponents of functions from RH~(～α,β)(B).